FRET - Förster Resonance Energy Transfer: From Theory to Applications

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In the biological sciences, FRET has been used to report on protein-protein interactions [ 11 , 12 ]. At the single-molecule level, FRET has been used to measure distances between labels in characterizing the structures and dynamics of macromolecules including RNA, DNA, proteins, and their molecular complexes [ 6 , 13 — 15 ].

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Time-depend FRET measurements have been developed to characterize reaction kinetics of enzymes [ 6 , 16 — 18 ], ligand-receptor interactions [ 7 , 19 — 21 ], conformational dynamics of proteins [ 13 , 22 , 23 ], and movement of molecular motor proteins [ 24 , 25 ]. Many types of molecules can be used for acceptor-donor pairing in FRET.

Some molecules have photophysics that result in non-emitting quenching when interacting with surrounding chemical species or intramolecular chemical groups [ 26 — 30 ]. In emerging biotechnology, FRET is also being used to develop new types of high-fidelity sensors for single-molecule detection and high-throughput assays for screening [ 7 , 8 , 20 ]. To characterize different molecular conformational states or the heterogeneous states of subpopulations, a ratiometric analysis is used to estimate the transfer efficiency E [ 18 , 32 ].

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Over repeated measurements this is reported typically as a histogram of the efficiency values E. Under differing experimental conditions, such as introduction of a denaturant, shifts in the observed efficiency histogram are interpreted as changes in the molecular conformational state [ 6 , 14 , 23 , 33 ].

In recent experiments by Lipman et al. This finding is supported by experiments where x-ray scattering of molecules indicate no conformational change or the molecular structure involved is inherently rigid such as a polyproline chain [ 34 , 35 ]. There is a precedent for such changes in efficiency occuring due to properties of the medium.

Experiments such as those by Zhang, Fu, Lakowicz, and others [ 36 , 37 ] demonstrate that the presence of foreign particles specifically silver in their studies can affect the donor-acceptor interaction. Furthermore, results by Makarov and Plaxco [ 38 ] suggest for a flexible polymer that not just the conformational state but the end-to-end kinetics can effect observed FRET efficiency. This presents the important issue of characterizing how shifts can occur in FRET efficiency in the apparent absence of any change in the conformational state.

We investigate using theory and stochastic simulations the roles played by excitation kinetics, orientation diffusion of fluorophores, separation diffusion of fluorophores, and non-emitting quenching. Our results aim to quantify the magnitude of these effects and to help identify regimes in which these factors could impact experimental measurements. The FRET efficiency is the fraction of energy that is transferred non-radiatively from the donor to the acceptor molecule.

Initially, it will be assumed that the energy can only be emitted as a donor photon or non-radiatively transferred to the acceptor ultimately to be emitted as an acceptor photon. The donor molecule is excited to a higher energy state by an adsorbed photon. The donor relaxes back to its ground state either by emitting a photon or transferring energy to the acceptor molecule.

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The excited state of the acceptor molecule relaxes by emitting photons. Shown are the two widely used donor-acceptor dyes Cy 3 and Cy 5. For some systems it may be important to consider also additional photo-chemical states as in [ 39 , 40 ] or transfer of energy from collisions with other molecules in solution that results in non-emitting quenching [ 28 — 30 ]. We consider some of these effects in subsequent sections.

FRET - Forster Resonance Energy Transfer

For a more detailed discussion see [ 1 — 3 , 5 , 10 , 42 ]. To obtain R 0 only requires in principle knowledge of a few properties of the photo-physics of the donor and acceptor molecules. This allows for FRET to be used as an effective nanoscale ruler for molecular systems [ 4 , 23 , 24 , 27 , 43 ].

To obtain single molecule measurements for freely diffusing molecules, the donor is typically excited by waiting for an individual molecule to diffuse into the focus of a laser beam [ 6 , 18 , 23 , 24 ]. When the molecule is in a region near enough to the focal point of the laser within the focal volume the donor is excited with high probability and a sequence of donor and acceptor photon emissions occur, see Fig 2.

During the time the molecule dwells in the focal volume, the number of detected donor and acceptor photons n D , n A can be counted. This allows for a ratio-metric estimate of the transfer efficiency as [ 18 , 32 ] 5 This experimental data for the FRET efficiency is then typically aggregated to form a histogram of the observed energy transfer efficiencies E.

We remark that there are a number of important considerations in practice for such experiments, such as the development of criteria for when such a sequence of emissions is to be considered a significant FRET event or when there are short durations in the focal volume or shot noise. A FRET event starts when a molecule labelled by a donor and acceptor pair diffuse into a volume of sufficiently large laser intensity near the focal point left. The counts for detected photon emissions for the acceptor n A and donor n D are recorded until the molecule diffuses out of the focal volume top right.

During the donor excitation either a photon is emitted or energy is non-radiatively transferred to the acceptor and emitted with rates that depend on the molecular conformation lower right. The efficiency histogram provides a characterization of the relative proportions of different conformational states or sub-populations of the molecules encountered during a measurement.

Introductory Concepts

For the case of homogeneous molecules in the same conformational state, the efficiency histogram is expected to exhibit a narrow peak around the characteristic FRET efficiency corresponding to the donar-acceptor separation of the conformation. It is then natural to consider changes in the conformational state of the molecule by looking for shifts in the location of the peak in the FRET histogram. This is widely used in experimental practice to characterize biomolecular systems [ 6 , 13 , 14 , 22 ].

However, in recent experiments by Lipman et al. We use theory and stochastic simulations to investigate the roles played by kinetics. We initially investigate the role played by the rotational and translational diffusion of fluorophores on the time-scale of the excitation kinetics of the donor and acceptor molecules.

We then consider the role of additional effects such as non-emitting quenching. We consider the role of the kinetics of donor and acceptor excitation, energy-transfer, and relaxation.

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This includes the separation distance R between the donor and acceptor. We investigate how such dependence of the energy transfer on the donor and acceptor configurations competes with the other excitation and relaxation kinetics. For this purpose we develop a stochastic model of the excitation-relaxation kinetics and perform simulations of the rotational and translational diffusion of the acceptor and donor molecules. The relative orientation of the dipole moments of the donor and acceptor molecules can significantly influence the efficiency of energy transfer [ 5 , 41 , 42 , 44 , 45 ].

The gives the separation unit vector pointing from the donor to acceptor. Contributions from orientation effects are often approximated by averaging assuming that the orientation rapidly diffuses isotropically on a time-scale much longer than the donor excitation time. However, in many situations the orientation diffusion can be comparable to the time-scale of excitations or from molecular-level sterics it may not be isotropic sampling all orientations [ 12 , 41 , 44 , 46 ]. This is sampled from a distribution with irregular and asymmetric features, see Fig 3.

The histogram was constructed from 10 7 random dye orientation pairs. We investigate the role of orientation diffusion and its role on observed FRET efficiencies leading to possible shifts. Since only the relative angle between the donor and acceptor is relevant, we can model rotational diffusion by a Brownian motion on the surface of a sphere [ 47 ].

The equations are to be interpreted in the sense of Ito Calculus [ 48 , 49 ]. The and denote independent Brownian motions. We perform simulations by numerically computing time-steps approximating the stochastic process in Eq 7. This is accomplished by projecting Brownian motion to the surface of the sphere. In particular, we use the time-stepping procedure 8 9 The is generated each step as a three-dimensional Gaussian random variable with independent components having mean zero and variance one.

We remark this approach avoids complications associated with the spherical coordinates by avoiding the need to switch coordinate charts when configurations approach the degeneracies near the poles of the sphere [ 50 ]. We consider the case when the acceptor and donor are free to rotate but are held at a fixed separation distance R.

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For slow rotational diffusion, we find that the limited sampling over the donor lifetime can result in significant shifts of the observed FRET transfer E toward lower efficiencies, see Fig 4. A notable feature for decreasing diffusivity is that the distribution of observed efficiencies broadens. As a consequence, the shift exhibited is a result of purely kinetic effects. In particular, for the fastest rotational diffusion the donor and acceptor have more opportunities to occupy orientations that are favourable to energy transfer.

When the rotational diffusion is much slower than the donor lifetime, the donor and acceptor orientation remain close to the initial starting configuration which primarily determines the rate of energy transfer. This is a consequence of the fast rotational diffusion having more opportunities to be in favourable orientations for energy transfer. The shift in transfer efficiencies resulting from the rotational kinetics can be significant. Our results indicate on way that the FRET transfer efficiency E can exhibit a significant shift without any change in the conformational state of the measured molecule.